Unit 6: Plane Kinematics and Kinetics of Rigid bodies

1. Introduction

In engineering mechanics, the study of rigid bodies in plane motion involves both kinematics (the geometry of motion without reference to the forces causing it) and kinetics (the relationship between the forces acting on a body and its resulting motion). A rigid body in plane motion undergoes translation, rotation, or a combination of both (general plane motion).

2. D'Alembert's Principle

2.1 Concept and Definition

D'Alembert's principle provides a method to transform a problem in dynamics into an equivalent problem in statics. It states that a system of rigid bodies in motion can be considered to be in a state of "dynamic equilibrium" by applying fictitious forces called Inertia Forces and Inertia Couples.

According to Newton's Second Law:

D'Alembert rearranged these equations to state:

• Inertia Force (): A fictitious force acting through the center of mass (G) in the direction opposite to the acceleration of the mass center.
• Inertia Couple (): A fictitious moment acting in the direction opposite to the angular acceleration of the body.

2.2 Application in Plane Motion

For a rigid body undergoing general plane motion, D'Alembert's principle allows us to draw a free-body diagram (FBD) that includes both the actual external forces/moments and the inertia forces/couples.

Steps for Analysis:

1. Establish a coordinate system (x, y).
2. Draw the Free Body Diagram (FBD) showing all external active and reactive forces.
3. Draw the Kinetic Diagram (or add inertia vectors to the FBD with dashed lines) showing:
   • Inertia force vector acting opposite to the assumed direction of .
   • Inertia couple acting opposite to the assumed direction of .
4. Apply the equations of dynamic equilibrium:
   
   
   

2.3 Application in Connected Bodies

When multiple rigid bodies are connected (e.g., via inextensible cords, pins, or linkages), their kinematics are dependent on one another.

• Kinematic relations: Establish the relationship between the accelerations of different bodies (e.g., if two blocks are connected by a rope over a pulley, and , and ).
• Isolate each body: Draw an FBD for each connected body.
• Apply D'Alembert's: Add the specific inertia forces/couples to each body based on its individual acceleration.
• Solve simultaneously: The internal forces (like tension in a rope) and unknown accelerations can be found by solving the dynamic equilibrium equations for all bodies simultaneously.

3. Work-Energy Principle

3.1 Concept and Definition

The Work-Energy Principle relates the work done by all external forces and moments on a rigid body to the change in its kinetic energy. It is an integral form of Newton's second law and is highly effective for problems involving displacements and velocities rather than accelerations and time.

Equation:

Where:

• = Initial kinetic energy
• = Final kinetic energy
• = Total work done by all external forces and moments during the displacement from state 1 to state 2.

3.2 Kinetic Energy of a Rigid Body ()

For a rigid body in general plane motion, the total kinetic energy is the sum of translational kinetic energy (of the center of mass, G) and rotational kinetic energy (about G).

• = velocity of the center of mass.
• = angular velocity of the rigid body.
• = mass moment of inertia about the center of mass.

Special Cases:

• Pure Translation: , so
• Pure Rotation about a fixed axis (O): . By the parallel axis theorem (), the kinetic energy simplifies to .

3.3 Work Done ()

• Work of a Force: . For a constant force, .
• Work of a Moment/Couple: . For a constant couple, .
• Work of Weight: (where is the change in elevation of the center of mass).
• Work of a Spring: (where is the deformation from the unstretched length).

3.4 Application in Plane Motion of Connected Bodies

For a system of connected rigid bodies, the principle can be applied to the system as a whole:

Key Considerations for Connected Bodies:

1. Internal Forces: Work done by internal forces (like tension in an inextensible cable connecting two masses) cancels out when considering the whole system.
2. Kinematic Constraints: Relate velocities and angular velocities of all components. For example, if a cable wraps around a pulley without slipping, .
3. Friction: Work done by kinetic friction is always negative (). In pure rolling without slipping, the static friction force does no work because the point of contact has zero velocity.

4. Kinetics of Rigid Body Rotation

When a rigid body is constrained to rotate about a fixed axis (e.g., a hinge or pivot point, let's call it point O), the motion of the center of mass (G) is a circular path around O.

4.1 Equations of Motion

For fixed-axis rotation, Newton's second law yields specific equations. Let the distance from the fixed axis O to the center of mass G be .

The acceleration of the center of mass has two components:

1. Tangential acceleration ():
2. Normal acceleration ():

Applying the equations of motion:

4.2 Moment Equation about the Fixed Axis

To avoid dealing with the unknown reaction forces at the pin/hinge O, it is often more convenient to take moments about the fixed axis O.
Using the parallel-axis theorem (), the moment equation simplifies dramatically:

Where:

• is the sum of moments of all external active forces about the fixed axis.
• is the mass moment of inertia of the body about the fixed axis.

4.3 Dynamic Reactions

In static equilibrium, the reactions at a pin support only balance the applied forces and weight. However, when the body is rotating, the reactions at the pin must also account for the inertia of the body (centripetal forces and tangential forces required to accelerate the mass center).

To find these dynamic reactions ( and ):

1. Determine and (using kinematics, energy methods, or the moment equation ).
2. Calculate the normal and tangential accelerations of the mass center: and .
3. Apply the force equations of motion ( and ) to solve for the unknown pin reactions.